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Math Help - Fibonacci Property Proof help

  1. #1
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    Question Fibonacci Property Proof help

    Hi,
    I was hoping that someone could help me with some proofs for the Fibonacci sequence.

    1) If p is prime then F_p \equiv \ \frac{5}{p} (mod p)
    2) If p is prime then  2F_{p+1} \equiv \ 1+ \frac{5}{p} (mod p)

    Thanks!
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  2. #2
    Super Member
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    Hmm?

    2 is prime. F2 = 1 \equiv 1 ~ (mod ~ 2)
    3 is prime. F3 = 1+1 = 2 \equiv 2 ~ (mod ~ 3)

    Are you sure this is the intended question?
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  3. #3
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    I took the equation directly from the sheet because I am having the same difficulties and currently just completely confused. I should have mentioned this in the first post but it is going off of the assumption that  F_1 = 1 , F_2 = 1 , F_3=2, F_4=3,..., F_n = F_{n-1} + F_{n-2} for  n \geq \ 3 So I don't know if that changes anything.
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  4. #4
    Super Member PaulRS's Avatar
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    Isn't it supposed to be: F_p\equiv{\left(\frac{5}{p}\right)_L}(\bmod.p) where \left(.\right)_L is the Legendre Symbol ?

    I give you the ingredients: Binet's Formula,the Binomial Theorem, Fermat's Little Theorem and Euler's Criterion.
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  5. #5
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    Yes I think you are right about it being a Legendre Symbol.
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